thumb|right|Examples of cyclic quadrilaterals

In geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertices all lie on a single circle, making the sides chords of the circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. The center of the circle and its radius are called the circumcenter and the circumradius respectively. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.

The word cyclic is from the Ancient Greek (kuklos), which means "circle" or "wheel".

All triangles have a circumcircle, but not all quadrilaterals do. An example of a quadrilateral that cannot be cyclic is a non-square rhombus. The section characterizations below states what necessary and sufficient conditions a quadrilateral must satisfy to have a circumcircle.

Special cases

Any square, rectangle, isosceles trapezoid, or antiparallelogram is cyclic. A kite is cyclic if and only if it has two right angles – a right kite. A bicentric quadrilateral is a cyclic quadrilateral that is also tangential and an ex-bicentric quadrilateral is a cyclic quadrilateral that is also ex-tangential. A harmonic quadrilateral is a cyclic quadrilateral in which the product of the lengths of opposite sides are equal.

Characterizations

thumb|upright=1.0|A cyclic quadrilateral ABCD with circumcenter M

Circumcenter

A convex quadrilateral is cyclic if and only if the four perpendicular bisectors to the sides are concurrent. This common point is the circumcenter.

Supplementary angles

thumb|[[Proof without words using the inscribed angle theorem that opposite angles of a cyclic quadrilateral are supplementary:<br />2𝜃 + 2𝜙 = 360° ∴ 𝜃 + 𝜙 = 180°]]

A convex quadrilateral is cyclic if and only if its opposite angles are supplementary, that is

<math display="block">\alpha + \gamma = \beta + \delta = \pi \ \text{radians}\ (= 180^{\circ}).</math>

The direct theorem was Proposition 22 in Book 3 of Euclid's Elements. Equivalently, a convex quadrilateral is cyclic if and only if each exterior angle is equal to the opposite interior angle.

In 1836 Duncan Gregory generalized this result as follows: Given any convex cyclic -gon, then the two sums of alternate interior angles are each equal to <math>(n-1)\pi</math>. This result can be further generalized as follows: lf is any cyclic -gon in which vertex (vertex is joined to ), then the two sums of alternate interior angles are each equal to (where and is the total turning).

Taking the stereographic projection (half-angle tangent) of each angle, this can be re-expressed,

<math display="block">\dfrac

{ \tan{\frac{\alpha}{2 + \tan{\frac{\gamma}{2 }

{ 1 - \tan{\frac{\alpha}{2 \tan{\frac{\gamma}{2 }

= \dfrac

{ \tan{\frac{\beta}{2 + \tan{\frac{\delta}{2 }

{ 1 - \tan{\frac{\beta}{2 \tan{\frac{\delta}{2 }

= \infty.</math>

Which implies that

<math display="block">\tan{\frac{\alpha}{2 \tan{\frac{\gamma}{2 = \tan{\frac{\beta}{2{\tan \frac{\delta}{2 = 1</math>

Angles between sides and diagonals

A convex quadrilateral is cyclic if and only if an angle between a side and a diagonal is equal to the angle between the opposite side and the other diagonal. That is, for example,

<math display="block">\angle ACB = \angle ADB.</math>

Pascal points

[[File:Cyclic_quadrilateral_-_pascal_points.png|thumb|ABCD is a cyclic quadrilateral. E is the point of intersection of the diagonals and F is the point of intersection of the extensions of sides BC and AD. <math>\omega</math> is a circle whose diameter is the segment, EF. P and Q are Pascal points formed by the circle <math>\omega</math>.

Triangles FAB and FCD are similar.]]

Other necessary and sufficient conditions for a convex quadrilateral to be cyclic are: let be the point of intersection of the diagonals, let be the intersection point of the extensions of the sides and , let <math>\omega</math> be a circle whose diameter is the segment, , and let and be Pascal points on sides and formed by the circle <math>\omega</math>.

<br> (1) is a cyclic quadrilateral if and only if points and are collinear with the center , of circle <math>\omega</math>.

<br> (2) is a cyclic quadrilateral if and only if points and are the midpoints of sides and .

<math display="block">\displaystyle AE\cdot EC = BE\cdot ED.</math>

The intersection may be internal or external to the circle. In the former case, the cyclic quadrilateral is , and in the latter case, the cyclic quadrilateral is . When the intersection is internal, the equality states that the product of the segment lengths into which divides one diagonal equals that of the other diagonal. This is known as the intersecting chords theorem since the diagonals of the cyclic quadrilateral are chords of the circumcircle.

Ptolemy's theorem

Ptolemy's theorem expresses the product of the lengths of the two diagonals and of a cyclic quadrilateral as equal to the sum of the products of opposite sides:

Four unequal lengths, each less than the sum of the other three, are the sides of each of three non-congruent cyclic quadrilaterals,

<math display="block">\displaystyle K=2R^2\sin{A}\sin{B}\sin{\theta}</math>

where is the radius of the circumcircle. As a direct consequence,

<math display="block">K\le 2R^2</math>

where there is equality if and only if the quadrilateral is a square.

Diagonals

In a cyclic quadrilateral with successive vertices , , , and sides , , , and , the lengths of the diagonals and can be expressed in terms of the sides as

<math display="block">p = \sqrt{\frac{(ac+bd)(ad+bc)}{ab+cd</math> and <math>q = \sqrt{\frac{(ac+bd)(ab+cd)}{ad+bc</math>

so showing Ptolemy's theorem

<math display="block">pq = ac+bd.</math>

According to Ptolemy's second theorem,

<math display="block">p+q\ge 2\sqrt{ac+bd}.</math>

Equality holds if and only if the diagonals have equal length, which can be proved using the AM-GM inequality.

Moreover,

<math display="block"> \frac{AE}{CE}=\frac{AB}{CB}\cdot\frac{AD}{CD}.</math>

A set of sides that can form a cyclic quadrilateral can be arranged in any of three distinct sequences each of which can form a cyclic quadrilateral of the same area in the same circumcircle (the areas being the same according to Brahmagupta's area formula). Any two of these cyclic quadrilaterals have one diagonal length in common.

<math display="block">\cos A = \frac{a^2-b^2-c^2+d^2}{2(ad+bc)},</math>

<math display="block">\sin A = \frac{2\sqrt{(s-a)(s-b)(s-c)(s-d){(ad+bc)},</math>

<math display="block">\tan \frac{A}{2} = \sqrt{\frac{(s-a)(s-d)}{(s-b)(s-c).</math>

The angle between the diagonals that is opposite sides and satisfies

A generalization of Mollweide's formula to cyclic quadrilaterals is given by the following two identities. Let <math>B</math> denote the angle between sides <math>a</math> and <math>b</math>, <math>C</math> the angle between <math>b</math> and <math>c</math>, and <math>D</math> the angle between <math>c</math> and <math>d</math>. If <math>E</math> is the point of intersection of the diagonals, denote <math>\angle{CED} = \theta</math>, then:

<math display="block">\begin{align}

\frac{a+c}{b+d}

&= \frac{\sin\tfrac12(A+B)}{\cos\tfrac12(C-D)}\tan\tfrac12\theta, \\[10mu]

\frac{a-c}{b-d}

&= \frac{\cos\tfrac12(A+B)}{\sin\tfrac12(D-C)}\cot\tfrac12\theta.

\end{align}</math>

Moreover, a generalization of the law of tangents for cyclic quadrilaterals is:

<math display="block">\frac{(a-c)(b-d)}{(a+c)(b+d)}=\frac{\tan\tfrac12(A-B)}{\tan\tfrac12(A+B)}.</math>

Parameshvara's circumradius formula

A cyclic quadrilateral with successive sides , , , and semiperimeter has the circumradius (the radius of the circumcircle) given by

<math display="block">R=\frac{1}{4} \sqrt{\frac{(ab+cd)(ac+bd)(ad+bc)}{(s-a)(s-b)(s-c)(s-d).</math>

This was derived by the Indian mathematician Vatasseri Parameshvara in the 15th century. (Note that the radius is invariant under the interchange of any side lengths.)

Using Brahmagupta's formula, Parameshvara's formula can be restated as

<math display="block">4KR=\sqrt{(ab+cd)(ac+bd)(ad+bc)}</math>

where is the area of the cyclic quadrilateral.

Anticenter and collinearities

Four line segments, each perpendicular to one side of a cyclic quadrilateral and passing through the opposite side's midpoint, are concurrent. which is an abbreviation for midpoint altitude. Their common point is called the anticenter. It has the property of being the reflection of the circumcenter in the "vertex centroid". Thus in a cyclic quadrilateral, the circumcenter, the "vertex centroid", and the anticenter are collinear.

If the diagonals of a cyclic quadrilateral intersect at , and the midpoints of the diagonals are and , then the anticenter of the quadrilateral is the orthocenter of triangle .

The anticenter of a cyclic quadrilateral is the Poncelet point of its vertices.

Other properties

thumb|right|Japanese theorem

  • In a cyclic quadrilateral , the incenters , , , (see the figure to the right) in triangles , , , and are the vertices of a rectangle. This is one of the theorems known as the Japanese theorem. The orthocenters of the same four triangles are the vertices of a quadrilateral congruent to , and the centroids in those four triangles are vertices of another cyclic quadrilateral.
  • If a cyclic quadrilateral has side lengths that form an arithmetic progression the quadrilateral is also ex-bicentric.
  • If the opposite sides of a cyclic quadrilateral are extended to meet at and , then the internal angle bisectors of the angles at and are perpendicular.

Brahmagupta quadrilaterals

A Brahmagupta quadrilateral is a cyclic quadrilateral with integer sides, integer diagonals, and integer area. It is a primitive Brahmagupta quadrilateral if no smaller geometrically similar quadrilateral has all these values as integers. Quadrilaterals whose side lengths, diagonals, and areas are all rational numbers are called rational Brahmagupta quadrilaterals in this article. Every primitive Brahmagupta quadrilateral is a rational Brahmagupta quadrilateral. Conversely, every rational Brahmagupta quadrilateral is geometrically similar to exactly one primitive Brahmagupta quadrilateral.

All primitive Brahmagupta quadrilaterals can be obtained from the following expressions involving rational parameters , , and . The computed side lengths , , , , diagonals , , area , and circumradius will be rational numbers. These can be scaled to produce a unique primitive Brahmagupta quadrilateral; note that is an area and will be scaled by the square of the value that multiplies the other quantities. The Brahmagupta quadrilateral will be non-self-intersecting and non-degenerate if .

<math display="block">a=[t(u+v)+(1-uv)][u+v-t(1-uv)]</math>

<math display="block">b=(1+u^2)(v-t)(1+tv)</math>

<math display="block">c=t(1+u^2)(1+v^2)</math>

<math display="block">d=(1+v^2)(u-t)(1+tu)</math>

<math display="block">e=u(1+t^2)(1+v^2)</math>

<math display="block">f=v(1+t^2)(1+u^2)</math>

<math display="block">K=uv[2t(1-uv)-(u+v)(1-t^2)][2(u+v)t+(1-uv)(1-t^2)]</math>

<math display="block">R=(1+u^2)(1+v^2)(1+t^2)/4.</math>

See for a different parameterization of all non-degenerate primitive Brahmagupta quadrilaterals, which depends upon rational numbers, .

Orthodiagonal case

Circumradius and area

For a cyclic quadrilateral that is also orthodiagonal (has perpendicular diagonals), suppose the intersection of the diagonals divides one diagonal into segments of lengths and and divides the other diagonal into segments of lengths and . Then (the first equality is Proposition 11 in Archimedes' Book of Lemmas)

<math display="block"> D^2=p_1^2+p_2^2+q_1^2+q_2^2=a^2+c^2=b^2+d^2 </math>

where is the diameter of the circumcircle. This holds because the diagonals are perpendicular chords of a circle. These equations imply that the circumradius can be expressed as

<math display="block"> R=\tfrac{1}{2}\sqrt{p_1^2+p_2^2+q_1^2+q_2^2} </math>

or, in terms of the sides of the quadrilateral, as

<math display="block"> K=\tfrac{1}{2}(ac+bd). </math>

Other properties

  • In a cyclic orthodiagonal quadrilateral, the anticenter coincides with the point where the diagonals intersect. One direction of this theorem was proved by Anders Johan Lexell in 1782. Lexell showed that in a spherical quadrilateral inscribed in a small circle of a sphere the sums of opposite angles are equal, and that in the circumscribed quadrilateral the sums of opposite sides are equal. The first of these theorems is the spherical analogue of a plane theorem, and the second theorem is its dual, that is, the result of interchanging great circles and their poles. Kiper et al. proved a converse of the theorem: If the summations of the opposite sides are equal in a spherical quadrilateral, then there exists an inscribing circle for this quadrilateral.

See also

  • Butterfly theorem
  • Brahmagupta triangle
  • Cyclic polygon
  • Power of a point
  • Ptolemy's table of chords
  • Robbins pentagon

References

Further reading

  • D. Fraivert: Pascal-points quadrilaterals inscribed in a cyclic quadrilateral
  • Derivation of Formula for the Area of Cyclic Quadrilateral
  • Incenters in Cyclic Quadrilateral at cut-the-knot
  • Four Concurrent Lines in a Cyclic Quadrilateral at cut-the-knot